Properties

Label 896.1.h.a
Level $896$
Weight $1$
Character orbit 896.h
Self dual yes
Analytic conductor $0.447$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -56
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 896.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.447162251319\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.1568.1
Artin image: $D_8$
Artin field: Galois closure of 8.0.1258815488.2

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} -\beta q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q -\beta q^{3} -\beta q^{5} - q^{7} + q^{9} + \beta q^{13} + 2 q^{15} + \beta q^{19} + \beta q^{21} + q^{25} + \beta q^{35} -2 q^{39} -\beta q^{45} + q^{49} -2 q^{57} + \beta q^{59} -\beta q^{61} - q^{63} -2 q^{65} + 2 q^{71} -\beta q^{75} + 2 q^{79} - q^{81} -\beta q^{83} -\beta q^{91} -2 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{7} + 2 q^{9} + 4 q^{15} + 2 q^{25} - 4 q^{39} + 2 q^{49} - 4 q^{57} - 2 q^{63} - 4 q^{65} + 4 q^{71} + 4 q^{79} - 2 q^{81} - 4 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
1.41421
−1.41421
0 −1.41421 0 −1.41421 0 −1.00000 0 1.00000 0
321.2 0 1.41421 0 1.41421 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.1.h.a 2
4.b odd 2 1 896.1.h.b yes 2
7.b odd 2 1 inner 896.1.h.a 2
8.b even 2 1 inner 896.1.h.a 2
8.d odd 2 1 896.1.h.b yes 2
16.e even 4 2 1792.1.c.d 2
16.f odd 4 2 1792.1.c.c 2
28.d even 2 1 896.1.h.b yes 2
56.e even 2 1 896.1.h.b yes 2
56.h odd 2 1 CM 896.1.h.a 2
112.j even 4 2 1792.1.c.c 2
112.l odd 4 2 1792.1.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.1.h.a 2 1.a even 1 1 trivial
896.1.h.a 2 7.b odd 2 1 inner
896.1.h.a 2 8.b even 2 1 inner
896.1.h.a 2 56.h odd 2 1 CM
896.1.h.b yes 2 4.b odd 2 1
896.1.h.b yes 2 8.d odd 2 1
896.1.h.b yes 2 28.d even 2 1
896.1.h.b yes 2 56.e even 2 1
1792.1.c.c 2 16.f odd 4 2
1792.1.c.c 2 112.j even 4 2
1792.1.c.d 2 16.e even 4 2
1792.1.c.d 2 112.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{71} - 2 \) acting on \(S_{1}^{\mathrm{new}}(896, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( -2 + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( -2 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( -2 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( -2 + T^{2} \)
$61$ \( -2 + T^{2} \)
$67$ \( T^{2} \)
$71$ \( ( -2 + T )^{2} \)
$73$ \( T^{2} \)
$79$ \( ( -2 + T )^{2} \)
$83$ \( -2 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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